We show that for any positive integer d, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cann...

The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both pol...

This paper addresses two strategies for stabilization of continuous time linear switched system. The first one, is of open loop nature (trajectory independent) and is based on the determination of a minimum dwell time by means of a family of quadratic Lyapunov functions. Interestingly, the proposed stability condition does not require the Lyapunov function be uniformly decreasing at every switc...

This paper proposes a hybrid control scheme for the synchronization of two chaotic Duffing oscillator system, subject to uncertainties and external disturbances. The novelty of this scheme is that the Linear Quadratic Regulation (LQR) control, Sliding Mode (SM) control and Gaussian Radial basis Function Neural Network (GRBFNN) control are combined to chaos synchronization with respect to extern...

In this paper we consider the class of discretetime systems switching between an arbitrary number p of autonomous positive subsystems. Necessary and sufficient conditions for the existence of (either linear or quadratic) copositive Lyapunov functions, whose values can be decreased in every positive state, by suitably choosing one of p subsystems, are obtained. When these conditions are fulfille...

We study the positive recurrence of piecewise Ornstein–Uhlenbeck (OU) diffusion processes, which arise from many-server queueing systems with phase-type service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to “overload” (service demand exceeds capacity) and “underload” (service capacity exceeds demand). The two regimes cause...

We consider the Lyapunov equation for the linear dynamics, which arises naturally when one seeks for a Lyapunov function with a uniform, exact decrease. In this setting, a solution to the Lyapunov equation has been characterized only for quadratic Lyapunov functions. We demonstrate that the Lyapunov equation is a well-posed equation for strictly stable dynamics and a much more general class of ...

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions—expressed as control barrier functions— to be unified with performance objectiv...

this paper deals with the problem of the state estimation of discrete time nonlinear systems described by Takagi-Sugeno structure where the premise variables are not measurable. The proposed observer design method is based on the use of the second method of Lyapunov and a quadratic function. Based on this procedure two methods are proposed. The convergence conditions of the observer are express...